A283799 Number of dispersed Dyck prefixes of length 2n and height n.

1, 2, 5, 12, 36, 90, 286, 728, 2380, 6120, 20349, 52668, 177100, 460460, 1560780, 4071600, 13884156, 36312408, 124403620, 326023280, 1121099408, 2942885946, 10150595910, 26681566392, 92263734836, 242799302200, 841392966470, 2216352204360, 7694644696200

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

Recursion: see Maple program.

a(n) = A282869(2n,n).

From Vaclav Kotesovec, Mar 26 2018: (Start)

Recurrence: 3*n*(3*n + 1)*(3*n + 2)*(3*n^3 - 11*n^2 + 10*n - 3)*a(n) = - 24*(2*n - 1)*(6*n^3 - 1)*a(n-1) + 64*(n-1)*(2*n - 3)*(2*n - 1)*(3*n^3 - 2*n^2 - 3*n - 1)*a(n-2).

a(n) ~ ((3+2*sqrt(3)) - (-1)^n*(3-2*sqrt(3))) * 2^(4*n + 1) / (sqrt(Pi*n) * 3^(3*n/2 + 2)). (End)

Maple

a:= proc(n) option remember; `if`(n<3, 1+n^2, ((512*(2*n-5))
      *(2519*n-1279)*(n-2)*(2*n-3)*a(n-3) +(192*(2*n-3))
      *(1710*n^3-443*n^2-4990*n+2483)*a(n-2) -(24*(22671*n^4
      -124866*n^3+216436*n^2-129032*n+24526))*a(n-1))
       / ((3*n+2)*(27*n+9)*(855*n-1504)*n))
    end:
seq(a(n), n=0..30);

Mathematica

b[x_, y_, m_] := b[x, y, m] = If[x == 0, z^m, If[y > 0, b[x - 1, y - 1, m], 0] + If[y == 0, b[x - 1, y, m], 0] + b[x - 1, y + 1, Max[m, y + 1]]];
a[n_] := Coefficient[b[2n, 0, 0], z, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz in A282869 *)

Crossrefs

Cf. A282869, A283667.

Sequence in context: A108555 A323397 A292169 * A225798 A303204 A333411

Adjacent sequences: A283796 A283797 A283798 * A283800 A283801 A283802

Keyword

nonn

Author

Alois P. Heinz, Mar 16 2017

Status

approved